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| Mirrors > Home > MPE Home > Th. List > Mathboxes > dfon2lem1 | Structured version Visualization version GIF version | ||
| Description: Lemma for dfon2 35793. (Contributed by Scott Fenton, 28-Feb-2011.) |
| Ref | Expression |
|---|---|
| dfon2lem1 | ⊢ Tr ∪ {𝑥 ∣ (𝜑 ∧ Tr 𝑥 ∧ 𝜓)} |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | truni 5275 | . 2 ⊢ (∀𝑦 ∈ {𝑥 ∣ (𝜑 ∧ Tr 𝑥 ∧ 𝜓)}Tr 𝑦 → Tr ∪ {𝑥 ∣ (𝜑 ∧ Tr 𝑥 ∧ 𝜓)}) | |
| 2 | nfsbc1v 3808 | . . . . 5 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑 | |
| 3 | nfv 1914 | . . . . 5 ⊢ Ⅎ𝑥Tr 𝑦 | |
| 4 | nfsbc1v 3808 | . . . . 5 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜓 | |
| 5 | 2, 3, 4 | nf3an 1901 | . . . 4 ⊢ Ⅎ𝑥([𝑦 / 𝑥]𝜑 ∧ Tr 𝑦 ∧ [𝑦 / 𝑥]𝜓) |
| 6 | vex 3484 | . . . 4 ⊢ 𝑦 ∈ V | |
| 7 | sbceq1a 3799 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑)) | |
| 8 | treq 5267 | . . . . 5 ⊢ (𝑥 = 𝑦 → (Tr 𝑥 ↔ Tr 𝑦)) | |
| 9 | sbceq1a 3799 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝜓 ↔ [𝑦 / 𝑥]𝜓)) | |
| 10 | 7, 8, 9 | 3anbi123d 1438 | . . . 4 ⊢ (𝑥 = 𝑦 → ((𝜑 ∧ Tr 𝑥 ∧ 𝜓) ↔ ([𝑦 / 𝑥]𝜑 ∧ Tr 𝑦 ∧ [𝑦 / 𝑥]𝜓))) |
| 11 | 5, 6, 10 | elabf 3675 | . . 3 ⊢ (𝑦 ∈ {𝑥 ∣ (𝜑 ∧ Tr 𝑥 ∧ 𝜓)} ↔ ([𝑦 / 𝑥]𝜑 ∧ Tr 𝑦 ∧ [𝑦 / 𝑥]𝜓)) |
| 12 | 11 | simp2bi 1147 | . 2 ⊢ (𝑦 ∈ {𝑥 ∣ (𝜑 ∧ Tr 𝑥 ∧ 𝜓)} → Tr 𝑦) |
| 13 | 1, 12 | mprg 3067 | 1 ⊢ Tr ∪ {𝑥 ∣ (𝜑 ∧ Tr 𝑥 ∧ 𝜓)} |
| Colors of variables: wff setvar class |
| Syntax hints: ∧ w3a 1087 ∈ wcel 2108 {cab 2714 [wsbc 3788 ∪ cuni 4907 Tr wtr 5259 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-v 3482 df-sbc 3789 df-ss 3968 df-uni 4908 df-iun 4993 df-tr 5260 |
| This theorem is referenced by: dfon2lem3 35786 dfon2lem7 35790 |
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